Cut bell peppers in half lengthwise, and discard seeds and membranes. Place pepper halves, skin sides up, on a foil-lined baking sheet; flatten with hand. Broil for 15 minutes or until blackened. Place in a zip-top plastic bag, and seal. Let stand for 15 minutes. Peel and cut into 1/2-inch strips. Combine bell peppers, oregano, 1 teaspoon oil, and 1/4 teaspoon salt.
Heat 1 teaspoon oil in a large nonstick skillet over medium heat. Add onion and 1/4 teaspoon salt; cook 15 minutes, stirring frequently. Increase heat to medium-high. Sprinkle onion with sugar; sauté 5 minutes or until lightly browned. Remove from pan.
Heat 1 teaspoon oil in pan over medium-high heat. Add mushrooms; cover and cook 2 minutes. Uncover and cook 1 minute or until tender, stirring constantly. Combine the onion, mushrooms, and 1/4 teaspoon salt.
Roll dough into a 12-inch circle. Place dough on grill rack coated with cooking spray; grill 3 minutes or until browned. Turn crust. Arrange the onion mixture and bell pepper mixture on crust. Top with cheese. Grill, covered, 3 minutes. Sprinkle with parsley and black pepper.